Van den Berg–Caffarelli–Lin hexagonal conjecture for minimal partitions

Let $\Omega$ be a bounded planar domain of area $A(\Omega)$, let $\mathfrak L_k(\Omega)$ denote the minimal energy of a $k$-partition of $\Omega$, and let $Hexa_1$ be the regular hexagon of area $1$. Van den Berg–Caffarelli–Lin hexagonal conjecture.

$$A(\Omega)\liminf_{k\rightarrow +\infty}\frac{\mathfrak L_k(\Omega)}{k}=A(\Omega)\limsup_{k\rightarrow +\infty}\frac{\mathfrak L_k(\Omega)}{k}=\lambda(Hexa_1).$$

Hexagonal tilings give the corresponding upper bound, but equality with the hexagonal value is not known in general; the conjecture predicts the asymptotic optimality of regular hexagonal cells.